<|"x"->{... big list ...},....|>

x:[xMinValue,xMaxValue] (where xMinValue=Min[{..Big List...}]...)

Format[f[x_?ListQ]] :=
 "f[x], x:[" <> ToString[Min[x]] <> ", " <> ToString[Max[x]] <> "]"

In[2]= f[Range[20]]]

Out[2]= f[x], x:[1, 20]

In[3]= Format[<|"x"->x_?ListQ|>] :=
 "x:[" <> ToString[Min[x]] <> ", " <> ToString[Max[x]] <> "]"

Unprotect[Association];
format[x_List] := "<< Very long list that we do not want to see, honestly. >>";
format[x_] := x;
Association /: Format[a_Association] :=
 Row[{"[LeftAssociation]",
   Sequence @@
    Riffle[Rule @@@ Transpose[{Keys[a], format /@ Values[a]}], ", "],
   "[RightAssociation]"}]
Protect[Association];

Association["Bli" -> 1, "Bla" -> Association["Blabla" -> {2, 3}], "Blubb" -> {4, 5}]

MyAssocationQ[assoc_] := MatchQ[assoc, <|"x" -> _?ListQ|>]

MyAssocationQ[<|"x" -> {-5, 2, 9}|>]

MyAssocationQ[2017]
MyAssocationQ[<|"x" -> {-5, 2, 9},"y"->1|>]

Unprotect[Association];

Association /: Format[assoc_Association /; MyAssocationQ[assoc]] :=
  "x:[" <> ToString[Min[assoc["x"]]] <> ", " <>
  ToString[Max[assoc["x"]]] <> "]";

Protect[Association];

In[]= <|"x" -> {-5, 2, 9}|>

In[]= <|"x" -> {-5, 2, 9}, "y" -> 5|>